1. Dissipate by heat, as in The sun will soon burn off the morning fog.
2. Clear land by burning vegetation, as in They've decided to burn off part of the field to prepare it for another planting. This practice has long been common in many parts of the world, but the precise term dates only from the first half of the 1800s.
[for some excess volatile or flammable substance] to burn away or burn up. A film of oil on the surface of the water was burning off, making dense black smoke. The alcohol burned off and left a delicious flavor in the cherries jubilee.